Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.0%

Time: 9.9s

Alternatives: 18

Speedup: 1.0×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y\_m \cdot y\_m, 0.0003968253968253968, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot \frac{-0.5 \cdot \sin x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 4e-11)
        (*
         (*
          (fma
           (fma
            (fma (* y_m y_m) 0.0003968253968253968 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         (/ (* -0.5 (sin x)) (- x)))
        (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 4e-11) {
		tmp = (fma(fma(fma((y_m * y_m), 0.0003968253968253968, 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * ((-0.5 * sin(x)) / -x);
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 4e-11)
		tmp = Float64(Float64(fma(fma(fma(Float64(y_m * y_m), 0.0003968253968253968, 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * Float64(Float64(-0.5 * sin(x)) / Float64(-x)));
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e-11], N[(N[(N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.0003968253968253968 + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(-0.5 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sin x \cdot \sinh y}}} \]

      3. frac-2neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\sin x \cdot \sinh y\right)}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh y \cdot \sin x}\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\sinh y \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \]

      8. lift-sinh.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\sinh y} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \]

      9. sinh-def N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{\left(\left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(x\right)} \cdot \color{blue}{\left(\left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\frac{-1}{x} \cdot \left(2 \cdot \sinh y\right)\right) \cdot \left(-0.5 \cdot \sin x\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(\frac{-1}{x} \cdot \color{blue}{\left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right)}\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \color{blue}{\left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right) \cdot y\right)}\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \color{blue}{\left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right) \cdot y\right)}\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right) + 2\right)} \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\left(\color{blue}{\left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right) \cdot {y}^{2}} + 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right), {y}^{2}, 2\right)} \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right) + \frac{1}{3}}, {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right) \cdot {y}^{2}} + \frac{1}{3}, {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}, {y}^{2}, \frac{1}{3}\right)}, {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {y}^{2} + \frac{1}{60}}, {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {y}^{2}, \frac{1}{60}\right)}, {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{y \cdot y}, \frac{1}{60}\right), {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{y \cdot y}, \frac{1}{60}\right), {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), \color{blue}{y \cdot y}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), \color{blue}{y \cdot y}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      15. unpow2 N/A

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), \color{blue}{y \cdot y}, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right) \]

      16. lower-*.f64 99.1

          \[\leadsto \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), \color{blue}{y \cdot y}, 2\right) \cdot y\right)\right) \cdot \left(-0.5 \cdot \sin x\right) \]

   7. Applied rewrites 99.1%

      \[\leadsto \left(\frac{-1}{x} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)}\right) \cdot \left(-0.5 \cdot \sin x\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot \left(\frac{-1}{2} \cdot \sin x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \sin x\right) \cdot \left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \sin x\right) \cdot \color{blue}{\left(\frac{-1}{x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{2} \cdot \sin x\right) \cdot \frac{-1}{x}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{2} \cdot \sin x\right) \cdot \frac{-1}{x}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{2} \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{x}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      7. frac-2neg N/A

         \[\leadsto \left(\left(\frac{-1}{2} \cdot \sin x\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(x\right)}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\frac{-1}{2} \cdot \sin x\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(x\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sin x}{\mathsf{neg}\left(x\right)}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sin x}{\mathsf{neg}\left(x\right)}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \sin x}}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{-1}{2}}}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{-1}{2}}}{\mathsf{neg}\left(x\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, y \cdot y, \frac{1}{60}\right), y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \]

      14. lower-neg.f64 99.3

          \[\leadsto \frac{\sin x \cdot -0.5}{\color{blue}{-x}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\sin x \cdot -0.5}{-x} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0003968253968253968, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 78.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.8%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0003968253968253968, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \frac{-0.5 \cdot \sin x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right)\\ t_1 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{\sin x}{x} \cdot t\_0\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
          (* y_m y_m)
          1.0))
        (t_1 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_1 (- INFINITY))
      (* (* t_0 (fma (* x x) -0.16666666666666666 1.0)) y_m)
      (if (<= t_1 4e-11) (* (* (/ (sin x) x) t_0) y_m) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0);
	double t_1 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_1 <= 4e-11) {
		tmp = ((sin(x) / x) * t_0) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0)
	t_1 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_1 <= 4e-11)
		tmp = Float64(Float64(Float64(sin(x) / x) * t_0) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, 4e-11], N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 78.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.8%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 4e-11)
        (* (* (fma (* y_m y_m) 0.16666666666666666 1.0) (/ (sin x) x)) y_m)
        (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 4e-11) {
		tmp = (fma((y_m * y_m), 0.16666666666666666, 1.0) * (sin(x) / x)) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 4e-11)
		tmp = Float64(Float64(fma(Float64(y_m * y_m), 0.16666666666666666, 1.0) * Float64(sin(x) / x)) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e-11], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 98.4

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right) \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}} + \frac{\sin x}{x}\right) \cdot y \]

      3. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({y}^{2} \cdot \frac{\sin x}{x}\right)} \cdot \frac{1}{6} + \frac{\sin x}{x}\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} + \frac{\sin x}{x}\right) \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)} + \frac{\sin x}{x}\right) \cdot y \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   8. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 78.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.8%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 1e-167)
        (*
         (pow
          (fma
           (fma 0.019444444444444445 (* x x) 0.16666666666666666)
           (* x x)
           1.0)
          -1.0)
         y_m)
        (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 1e-167) {
		tmp = pow(fma(fma(0.019444444444444445, (x * x), 0.16666666666666666), (x * x), 1.0), -1.0) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 1e-167)
		tmp = Float64((fma(fma(0.019444444444444445, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) ^ -1.0) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-167], N[(N[Power[N[(N[(0.019444444444444445 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-167

   1. Initial program 72.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.2

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-167 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 64.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.2%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 1e-167)
        (*
         (pow
          (fma
           (fma 0.019444444444444445 (* x x) 0.16666666666666666)
           (* x x)
           1.0)
          -1.0)
         y_m)
        (*
         0.5
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 1e-167) {
		tmp = pow(fma(fma(0.019444444444444445, (x * x), 0.16666666666666666), (x * x), 1.0), -1.0) * y_m;
	} else {
		tmp = 0.5 * (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 1e-167)
		tmp = Float64((fma(fma(0.019444444444444445, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) ^ -1.0) * y_m);
	else
		tmp = Float64(0.5 * Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-167], N[(N[Power[N[(N[(0.019444444444444445 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[(0.5 * N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-167

   1. Initial program 72.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.2

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-167 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 64.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (*
       (*
        (fma 0.16666666666666666 (* y_m y_m) 1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 1e-167)
        (*
         (pow
          (fma
           (fma 0.019444444444444445 (* x x) 0.16666666666666666)
           (* x x)
           1.0)
          -1.0)
         y_m)
        (*
         0.5
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = (fma(0.16666666666666666, (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 1e-167) {
		tmp = pow(fma(fma(0.019444444444444445, (x * x), 0.16666666666666666), (x * x), 1.0), -1.0) * y_m;
	} else {
		tmp = 0.5 * (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 1e-167)
		tmp = Float64((fma(fma(0.019444444444444445, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) ^ -1.0) * y_m);
	else
		tmp = Float64(0.5 * Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-167], N[(N[Power[N[(N[(0.019444444444444445 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[(0.5 * N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 62.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-167

   1. Initial program 72.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.2

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-167 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 64.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right) \cdot y\_m, y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (*
       (*
        (fma 0.16666666666666666 (* y_m y_m) 1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 1e-167)
        (*
         (pow
          (fma
           (fma 0.019444444444444445 (* x x) 0.16666666666666666)
           (* x x)
           1.0)
          -1.0)
         y_m)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666) y_m)
          y_m
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = (fma(0.16666666666666666, (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 1e-167) {
		tmp = pow(fma(fma(0.019444444444444445, (x * x), 0.16666666666666666), (x * x), 1.0), -1.0) * y_m;
	} else {
		tmp = fma((fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 1e-167)
		tmp = Float64((fma(fma(0.019444444444444445, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.0) ^ -1.0) * y_m);
	else
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-167], N[(N[Power[N[(N[(0.019444444444444445 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 62.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-167

   1. Initial program 72.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.2

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-167 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 67.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 67.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-167}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right) \cdot y\_m, y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (*
       (*
        (fma 0.16666666666666666 (* y_m y_m) 1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 4e-11)
        (* (pow (fma (* x x) 0.16666666666666666 1.0) -1.0) y_m)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666) y_m)
          y_m
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = (fma(0.16666666666666666, (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 4e-11) {
		tmp = pow(fma((x * x), 0.16666666666666666, 1.0), -1.0) * y_m;
	} else {
		tmp = fma((fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 4e-11)
		tmp = Float64((fma(Float64(x * x), 0.16666666666666666, 1.0) ^ -1.0) * y_m);
	else
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e-11], N[(N[Power[N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 62.4%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 76.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot {x}^{2}} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)} \cdot y \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 67.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\right)}^{-1} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right) \cdot y\_m, y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -1e-147)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 4e-11)
        (* (pow (fma (* x x) 0.16666666666666666 1.0) -1.0) y_m)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666) y_m)
          y_m
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -1e-147) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 4e-11) {
		tmp = pow(fma((x * x), 0.16666666666666666, 1.0), -1.0) * y_m;
	} else {
		tmp = fma((fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -1e-147)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 4e-11)
		tmp = Float64((fma(Float64(x * x), 0.16666666666666666, 1.0) ^ -1.0) * y_m);
	else
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-147], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e-11], N[(N[Power[N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999997e-148

   1. Initial program 99.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 25.4

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 32.0%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 32.0%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -9.9999999999999997e-148 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 76.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\frac{x}{\sin x}} \cdot y \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot {x}^{2}} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)} \cdot y \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 67.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\right)}^{-1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma (* x x) -0.16666666666666666 1.0))
       y_m)
      (if (<= t_0 4e-11) (* (/ (sin x) x) y_m) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma((x * x), -0.16666666666666666, 1.0)) * y_m;
	} else if (t_0 <= 4e-11) {
		tmp = (sin(x) / x) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)) * y_m);
	elseif (t_0 <= 4e-11)
		tmp = Float64(Float64(sin(x) / x) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e-11], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.99999999999999976e-11

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 98.4

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   if 3.99999999999999976e-11 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \cdot \frac{1}{2} \]

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 78.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.8%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\_m\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right) \cdot y\_m, y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -2e-213)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 5e-318)
        (* (* (/ -1.0 x) y_m) (- x))
        (*
         (fma
          (* (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666) y_m)
          y_m
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 5e-318) {
		tmp = ((-1.0 / x) * y_m) * -x;
	} else {
		tmp = fma((fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -2e-213)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 5e-318)
		tmp = Float64(Float64(Float64(-1.0 / x) * y_m) * Float64(-x));
	else
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666) * y_m), y_m, 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-213], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-318], N[(N[(N[(-1.0 / x), $MachinePrecision] * y$95$m), $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 35.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999987e-318

   1. Initial program 63.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.1

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.9%

      \[\leadsto \left(-\sin x\right) \cdot \color{blue}{\left(\frac{-1}{x} \cdot y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 88.0%

       \[\leadsto \left(-x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]

   if 4.9999987e-318 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\_m\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y\_m \cdot y\_m\right) \cdot 0.008333333333333333, y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -2e-213)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 5e-318)
        (* (* (/ -1.0 x) y_m) (- x))
        (* (fma (* (* y_m y_m) 0.008333333333333333) (* y_m y_m) 1.0) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 5e-318) {
		tmp = ((-1.0 / x) * y_m) * -x;
	} else {
		tmp = fma(((y_m * y_m) * 0.008333333333333333), (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -2e-213)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 5e-318)
		tmp = Float64(Float64(Float64(-1.0 / x) * y_m) * Float64(-x));
	else
		tmp = Float64(fma(Float64(Float64(y_m * y_m) * 0.008333333333333333), Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-213], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-318], N[(N[(N[(-1.0 / x), $MachinePrecision] * y$95$m), $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 35.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999987e-318

   1. Initial program 63.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.1

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.9%

      \[\leadsto \left(-\sin x\right) \cdot \color{blue}{\left(\frac{-1}{x} \cdot y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 88.0%

       \[\leadsto \left(-x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]

   if 4.9999987e-318 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\_m\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 -2e-213)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 5e-318)
        (* (* (/ -1.0 x) y_m) (- x))
        (* (fma 0.16666666666666666 (* y_m y_m) 1.0) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 5e-318) {
		tmp = ((-1.0 / x) * y_m) * -x;
	} else {
		tmp = fma(0.16666666666666666, (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= -2e-213)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 5e-318)
		tmp = Float64(Float64(Float64(-1.0 / x) * y_m) * Float64(-x));
	else
		tmp = Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-213], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-318], N[(N[(N[(-1.0 / x), $MachinePrecision] * y$95$m), $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 35.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999987e-318

   1. Initial program 63.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.1

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.9%

      \[\leadsto \left(-\sin x\right) \cdot \color{blue}{\left(\frac{-1}{x} \cdot y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 88.0%

       \[\leadsto \left(-x\right) \cdot \left(\color{blue}{\frac{-1}{x}} \cdot y\right) \]

   if 4.9999987e-318 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\left(\frac{-1}{x} \cdot y\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -2e-213)
    (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
    (* (fma 0.16666666666666666 (* y_m y_m) 1.0) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -2e-213) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else {
		tmp = fma(0.16666666666666666, (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -2e-213)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	else
		tmp = Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 35.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 55.1%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -2e-213)
    (* (* -0.16666666666666666 (* x x)) y_m)
    (* (fma 0.16666666666666666 (* y_m y_m) 1.0) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -2e-213) {
		tmp = (-0.16666666666666666 * (x * x)) * y_m;
	} else {
		tmp = fma(0.16666666666666666, (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -2e-213)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * x)) * y_m);
	else
		tmp = Float64(fma(0.16666666666666666, Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 12.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 55.1%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -2e-213)
    (* (* -0.16666666666666666 (* x x)) y_m)
    (* 1.0 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -2e-213) {
		tmp = (-0.16666666666666666 * (x * x)) * y_m;
	} else {
		tmp = 1.0 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= (-2d-213)) then
        tmp = ((-0.16666666666666666d0) * (x * x)) * y_m
    else
        tmp = 1.0d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= -2e-213) {
		tmp = (-0.16666666666666666 * (x * x)) * y_m;
	} else {
		tmp = 1.0 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= -2e-213:
		tmp = (-0.16666666666666666 * (x * x)) * y_m
	else:
		tmp = 1.0 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -2e-213)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * x)) * y_m);
	else
		tmp = Float64(1.0 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= -2e-213)
		tmp = (-0.16666666666666666 * (x * x)) * y_m;
	else
		tmp = 1.0 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(1.0 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 33.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 12.9%

       \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y \]

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 60.0

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 32.0%

      \[\leadsto 1 \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\sin x \cdot \frac{\sinh y\_m}{x}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* (sin x) (/ (sinh y_m) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (sin(x) * (sinh(y_m) / x));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * (sin(x) * (sinh(y_m) / x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * (Math.sin(x) * (Math.sinh(y_m) / x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * (math.sin(x) * (math.sinh(y_m) / x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(sin(x) * Float64(sinh(y_m) / x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * (sin(x) * (sinh(y_m) / x));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y$95$m], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sinh y}}{x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   6. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 18: 27.9% accurate, 36.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(1 \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* 1.0 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (1.0 * y_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * (1.0d0 * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * (1.0 * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * (1.0 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(1.0 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * (1.0 * y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(1.0 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

   5. lower-sin.f64 50.1

      \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 29.7%

   \[\leadsto 1 \cdot y \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024271 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))